Look carefully at the links provided by Leroy.
Apple does say this: "If additional cleaning is required, slightly dampen the cloth with water or a cleaner intended for use with a screen or display, then wipe the screen. Avoid getting moisture in openings."
Also this: "To clean hard-to-remove smudges or fingerprints on the display or exterior of your Mac, you can use a cloth moistened with a 70-percent isopropyl alcohol (IPA) solution to gently wipe the display or enclosure of your Mac laptop."
It looks like you have a very oily residue on your screen. This might not clean well with just a damp cloth. (An aside -- in the future, perhaps you should require any users to thoroughly wash and dry their hands before using the laptop.) See the above quoted excerpts from Apple's online documentation. Beyond water, Apple suggests careful use with a "cleaner intended for use with a screen" and "70% isopropyl alcohol." I would first carefully test any non-water cleaner first in a corner or extreme edge of the screen to verify expected results before using across the entire screen.